## Radius and Density of Atomic Nucleus

**Typical nuclear radii** are of the order **10**^{−14}** m**. Assuming spherical shape, nuclear radii can be calculated according to following formula:

r = r_{0} . A^{1/3}

where r_{0} = 1.2 x 10^{-15 }m = 1.2 fm

If we use this approximation, we therefore expect the **geometrical cross-sections** of nuclei to be of the order of πr^{2} or **4.5×10**^{−30 }**m² for hydrogen** nuclei or **1.74×10**^{−28}** m² for **^{238}**U** nuclei.

**Nuclear density** is the density of the nucleus of an atom. It is the ratio of mass per unit volume inside the nucleus. Since atomic nucleus carries most of atom’s mass and atomic nucleus is very small in comparison to entire atom, the nuclear density is very high.

The nuclear density for a typical nucleus can be approximately calculated from the size of the nucleus and from its mass. For example, **natural uranium** consists primarily of isotope ^{238}U (99.28%), therefore the atomic mass of uranium element is close to the atomic mass of ^{238}U isotope (238.03u). Its radius of this nucleus will be:

r = r_{0} . A^{1/3} = 7.44 fm.

Assuming it is spherical, its volume will be:

V = 4πr^{3}/3 = 1.73 x 10^{-42} m^{3}.

The usual definition of nuclear density gives for its density:

ρ_{nucleus} = m / V = 238 x 1.66 x 10^{-27} / (1.73 x 10^{-42}) = **2.3 x 10**^{17}**kg/m**** ^{3}**.

Thus, the density of nuclear material is more than 2.10^{14} times greater than that of water. It is an immense density. The descriptive term *nuclear density* is also applied to situations where similarly high densities occur, such as within neutron stars. Such immense densities are also found in neutron stars.

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